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A377011
Table read by rows: row n is the unique primitive Pythagorean triple whose short leg is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
7
1, 0, 1, 7, 24, 25, 41, 840, 841, 239, 28560, 28561, 1393, 970224, 970225, 8119, 32959080, 32959081, 47321, 1119638520, 1119638521, 275807, 38034750624, 38034750625, 1607521, 1292061882720, 1292061882721, 9369319, 43892069261880, 43892069261881, 54608393, 1491038293021224, 1491038293021225
OFFSET
0,4
COMMENTS
A002315(n) is a NSW number (named after Newman, Shanks, and Williams) and is an integer h that solves the Diophantine equation 2*n^2 = h^2 + 1.
Row n=0 is included by convention and corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.
The hypotenuses of all these Pythagorean triples are perfect squares.
REFERENCES
Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
FORMULA
row(n) = (A002315(n), A078522(n+1), A008844(n)).
row(n) = (2*A114336(3n-2) + 1, (A114336(3n))^2 - 1, (A114336(3n))^2) for all n > 0.
EXAMPLE
Table begins:
n=0: 1, 0, 1;
n=1: 7, 24, 25;
n=2: 41, 840, 841;
n=3: 239, 28560, 28561;
n=4: 1393, 970224, 970225;
n=5: 8119, 32959080, 32959081;
MATHEMATICA
t[n_]:=t[n]= Module[{a, b}, a=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2; b=(a^2-1)/2; {a, b, b+1}]; ternas={}; Do[ternas=Join[ternas, FullSimplify[t[n]]], {n, 0, 10}]; ternas
CROSSREFS
Cf. A002315 (short leg), A078522 (long leg), A008844 (hypotenuse), A001652 (inradius), A377016 (semiperimeter), A377017 (area), A001653 (square root of the hypotenuse), A114336.
Sequence in context: A286506 A286406 A070410 * A077035 A076602 A287132
KEYWORD
nonn,easy,tabf
STATUS
approved